October 2024 National Unified Academic Achievement Test (Grade 12) — Geometry

What is the value of ?

For the function , what is the value of ?

For where and , what is the value of ?

What is the value of ?

For the function to be continuous on the set of all real numbers, what is the sum of all constant values of ?

Let be the sum of the first terms of a geometric sequence with positive common ratio. Given that find the value of .

When a line passing through the point with slope bisects the area of the region enclosed by the graph of the function , the -axis, and the two lines and , what is the value of the constant ?

On the coordinate plane, there are two points and . When the point that externally divides the line segment in the ratio lies on the line , what is the value of the positive number ?

At time , two points and simultaneously start from the origin and move along a number line. Their velocities at time are respectivelyAfter starting, for a positive number such that the two points and meet exactly once, what is the distance traveled by point from time to time ?

As shown in the figure, for quadrilateral ABCD inscribed in a circle with Let , , then . If the intersection point of line segments AC and BD is E, what is the length of line segment AE? (Given that )

For a quartic function with leading coefficient , the function is differentiable on the entire set of real numbers, and the equation of the tangent line at point on the curve is . What is the sum of all distinct real values of such that ?

For a sequence where all terms are natural numbers, satisfying the following condition for all natural numbers : Find the sum of all values of such that .

Find the value of the real number that satisfies the equation .

For the function , find the value of .

For a sequence and constant , given that find the value of .

For two natural numbers and , the function is defined as: For a real number , let be the number of distinct real roots of the equation with respect to . When the function satisfies the following conditions, find the minimum value of . (a) The range of function is . (b) The number of natural numbers for which is .

What is the distance between the two foci of the hyperbola ?

Let point in coordinate space be reflected across the -plane to get point . For point , when the point that divides line segment in the ratio lies on the -axis, what is the value of ?

For two vectors , Find the value of .

A line passing through the focus F of the parabola with positive slope intersects the parabola at two points A and B respectively. When , what is the y-intercept of the tangent line at point A on this parabola?

In a cube ABCD-EFGH with edge length 2 as shown in the figure, let M be the midpoint of edge DH and N be the midpoint of edge GH. For a point P on line segment FM such that the length of line segment NP is minimized, what is the length of the orthogonal projection of line segment NP onto plane FHM?

For two points A(9,0) and B(8,1) on the coordinate plane, let S be the set of all points X that satisfy the following conditions. (a) (b) There exists a real number such that . Let P be the point in set S with the maximum x-coordinate. When the angle between vectors and is , what is the value of ? (Here, O is the origin.)

Let be an ellipse with major axis length and two foci . Let be an ellipse with major axis length and two foci , . Let be the intersection point of the two ellipses and with positive -coordinate. When , , form an arithmetic sequence in this order, . Find the value of . (where are integers.)

As shown in the figure, there is a square pyramid A-BCDE with a square base having side length 2 and . Let P and Q be the midpoints of line segments BC and CD respectively, and let R be the point that divides line segment CA in the ratio 1:7. Among the points on the sphere passing through all four points C, P, Q, and R, let S be the point where the distance to line AB is minimized. When the area of the orthogonal projection of triangle ABS onto plane BCD is , find the value of . (Here, p and q are rational numbers.)